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A Necessary and Sufficient Condition for the Equivalence of the Topologies of Uniform and Compact Convergence

Published online by Cambridge University Press:  20 November 2018

Michael H. Clapp  and
Ray C. Shiflett
Michael H. Clapp
Affiliation:
Department of Mathematics, California State Univ. Fullerton, Fullerton, California 92634
Ray C. Shiflett
Affiliation:
Department of Mathematics, California State Univ. Fullerton, Fullerton, California 92634
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Let (X, p) and (Y, d) be metric spaces with at least two points. It is usual for introductory courses in topology to study the set Yx of all functions mapping X to Y with the pointwise, compact-open, uniform convergence, and uniform convergence on compacta topologies. Some care is taken to show sufficient conditions for these topologies to be equivalent [1, 2]. However, the question of necessary conditions are dismissed with examples showing that the topologies are not in general equivalent.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 22 , Issue 4 , 01 December 1979 , pp. 467 - 470
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Dugundji, James, Topology, Allyn and Bacon, 1966.Google Scholar
2. Pervin, William, Foundations of General Topology, Academic Press, 1964.Google Scholar